The effects of temperature on current-potential profiles from electrochemical film production: A theoretical approach

Article abstract of DOI:10.1016/j.electacta.2007.11.035

In this paper we improve an equation that describes the current-potential curves obtained during electrochemical deposition of metals on n-silicon. The equation give us the theoretical description of the voltammograms and was previously introduced in terms of the ion concentration and the potential, however it still requires the inclusion of factors that describe the influence of the temperature. Temperature controls the diffusion constant D, the electrical resistivity of the electrolytic solution ρ, and the conduction electron density on the electrode surface N. In this paper we take explicitly into account the dependence of D, ρ and N on the temperature and succeeded to relate them to a defined reaction rate k. To complete the description, we considered that the influence of the temperature could be accounted by renormalizing the magnitude of the potential that triggers the deposition. Thus, a final expression for the current I, as a function of voltage V, ion concentration c_b and temperature T is achieved and a qualitative comparison between theoretical and experimental data is made. Through the comparisons, we show that the temperature affects the magnitude of the stationary currents, the amplitude of the nucleation loops and the intensity of diffusion-limited growth peak, producing a shift of the current-voltage curves toward less negative values of V. The same description allowed us to observe similar effects in the current transients.

Full text of DOI:10.1016/j.electacta.2007.11.035

Available online at www.sciencedirect.com
Electrochimica Acta 53 (2008) 3156–3165
The effects of temperature on current–potential profiles from
electrochemical film production: A theoretical approach
P.C.T. D’Ajello∗, A.A. Pasa, M.L. Munford, A.Q. Schervenski
Universidade Federal de Santa Catarina, Departamento de F´ısica/CFM, PO Box 476, CEP 88040-900, Brazil
Received 21 August 2007; received in revised form 9 November 2007; accepted 14 November 2007
Available online 22 November 2007
Abstract
In this paper we improve an equation that describes the current–potential curves obtained during electrochemical deposition of metals on n-
silicon. The equation give us the theoretical description of the voltammograms and was previously introduced in terms of the ion concentration
and the potential, however it still requires the inclusion of factors that describe the influence of the temperature. Temperature controls the diffusion
constant D, the electrical resistivity of the electrolytic solution ρ, and the conduction electron density on the electrode surface N. In this paper
we take explicitly into account the dependence of D, ρ and N on the temperature and succeeded to relate them to a defined reaction rate k. To
complete the description, we considered that the influence of the temperature could be accounted by renormalizing the magnitude of the potential
that triggers the deposition. Thus, a final expression for the current I, as a function of voltage V, ion concentration cb and temperature T is achieved
and a qualitative comparison between theoretical and experimental data is made. Through the comparisons, we show that the temperature affects
the magnitude of the stationary currents, the amplitude of the nucleation loops and the intensity of diffusion-limited growth peak, producing a shift
of the current–voltage curves toward less negative values of V. The same description allowed us to observe similar effects in the current transients.
© 2007 Elsevier Ltd. All rights reserved.
Keywords: Voltammograms; Current transients; Diffusion; Reaction; Electrochemical deposition
1. Introduction
to describe the current transients and despite the high quality of
all these models, it still lacks current expressions put forward
as a function of those parameters, currently used to control the
deposition processes.
Electrochemistry has been studied, tested and explored over
the last 200 years without loosing interest by the academic com-
munity. In fact its appeal grows renewed every day because
the claim for new technologies, related to microelectronic
applications [1–4]. Moreover many biological processes, from
biological sensors to ion transport through membranes [5], seem
to require for a deep understanding on electrochemical depo-
sition. However the application of electrochemical techniques
to other fields demands for theoretical descriptions that made
explicit a relation between the transients and the many physical
variables that renders singular every system, that is, temper-
ature, potential, ion concentration, electrical resistivity of the
electrolytic bath, and time.
As a consequence, in this article the main purpose is to
improve a former theoretical description of the transients and
voltammograms [26,27], putting them as functions of these
parameters, namely: temperature, potential, ion concentration,
electrical resistivity of the electrolytic bath, and time.
In the first article [26], we derived and expression for the
current transients assuming that there are two fundamental
mechanisms to be considered, namely the diffusion of ions near
the electrode and the reactions on the electrode surface. Fick’s
second law,
∂
∂
∂x2
From the beautiful review given by Hyde and Compton [6] we
realize that notwithstanding the many models [7–25] proposed
c(x, t) = D
c(x, t),
(1)
∂t
was solved assuming a finite system regulated by a time-
dependent boundary condition,
∗
Corresponding author.
c(0, t) = (cb − cs) exp(−kt) + cs,
(2a)
E-mail address: pcesar@fisica.ufsc.br (P.C.T. D’Ajello).
0013-4686/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.electacta.2007.11.035

P.C.T. D’Ajello et al. / Electrochimica Acta 53 (2008) 3156–3165
3157
that takes into account the reaction rate k, i.e. rate at which the
ions are reduced at the surface and two boundary conditions:
In a voltammogram, Vk demarks the onset potential for ion
reduction or the potential at which the reaction stop, when the
applied potential retraces back to the initial value.
c(x, 0) = cb, 0 < x < δ
c(δ, t) = cb, ∀t.
(2b)
(2c)
Because V evolves in time through a constant rate ω (the
scan rate), it is written through the linear equation V = V0 + ωt,
where V0 is the starting applied potential. As a consequence, t
and V are connected variables and Eq. (3) could be used to depict
the current transient curves I(cb, V, T, t), when concentration,
potential and temperature were fixed or, yet, to reproduce the
voltammograms I(cb, ω, T, V) at fixed concentration, scan rate
and temperature.
In Eqs. (1) and (2) c(x, t) is the ion’s concentration, D the dif-
fusion constant, cb and cs the ion’s concentration in the bulk and
on the electrode surface, respectively. δ is a length that defines
the thickness of the stationary diffusion layer, in which the ion
concentration decay.
In order to make a complete description, we will open the cur-
rentdependencyontemperatureintroducingitintotheparameter
␣ of Eq. (4), which has a dependency on the diffusion constant D,
the electrical resistivity ρ and conduction electron density at the
surface of the electrode N. To conclude the article we perform
a qualitative comparison between theoretical and experimental
voltammograms to explore the plasticity of the model.
Under conditions (2) we solve Eq. (1) to obtain the current
density expression, given by [26]:
zFD(cb − cs)
I(t) = −
δ
ꢀ
ꢂ
∞
2
ꢁ
exp(−kt) − exp(−λi t)
×
1 − exp(−kt) + 2k
,
λ2i − k
i=1
(3)
2. The reaction rate k as function of temperature
In Eq. (3) z is the charge number, F the Faraday constant and
λi = (iπ/δ)2D is a parameter that appears during the solution of
Fick’s second equation, an eigenvalue.
Once the current density, Eq. (3), is given, any additional
physical information must be included in the reaction rate k.
As a consequence in a second article [27], we made explicit the
connection between the reaction rate k and the potential, through
the formula,
In order to put forward and expression for the reaction rate k
in terms of the temperature, we will follow a course that starts
with the identification of all factors (variables and parameters)
affected by the thermal conditions observed in the electrolytic
cell. Thenextstepwillbetofindaconvenientrelationshipamong
these factors; such that the final expression for the currents
attends all features present in the experimental curves obtained
at different temperatures.
To define the effect of temperature on the system, we must
recall that we assumed that the electrochemical deposition pro-
cess evolves through a two-step type mechanism [26,27]. The
ions diffuse towards the electrode surface to be incorporated
by the growing deposit by reaction that has a probability to
occur at an electro-active point on the surface. The temperature
through different parameters influences both these mechanisms,
diffusion and reaction.
While it is easy to state that diffusion is sensitive to a change
in temperature through the magnitude of its transport coefficient,
it is not so simple to discriminate the effects produced by tem-
perature on the charge transfer reaction. However, because this
reaction is strongly determined by the available charge to be
transferred at the electrode/electrolyte interface, we have cho-
sen the population of conduction band electrons, on the electrode
surface, as a relevant parameter. At this interface the electrical
resistivity of the electrolytic solution must be considered as one
of the factors that affects the charge transfer rates. Finally, we
realize a last effect that should affect the reaction rate. Because of
charge transfer reactions are conditioned by an activation energy,
which represents the minimum amount of energy necessary to
the electrons to surmount the energy barrier for reduction of the
ions, a thermal contribution from the bath should be included as
aiding the activation energy. That is, the bath temperature also
influences the barrier height for electron transfer at the electrode
surface, affecting the potential onset of the reduction reactions.
Assuming that these are the relevant factors related to tem-
perature that produce significant changes in the reaction rate, we
ꢃ
ꢄ
ꢅꢆ−1
zFα(V − Vl)
k = [1 + exp{b(V − Vk)}]−1 1 + exp
.
RT
(4)
This expression was derived on the assumption that an ion on
the electrode surface must choose one of two possible outcomes:
to receive or not an electron charge from the electrode, that is,
even when there are afforded advantage for a charge transfer
there is a finite probability for the non-occurrence of a reduction
reaction.
In Eq. (4) R is the gas constant and ␣ is a non-dimensional
factor that must contain the contributions that came from another
physical condition, other than those related to a potential differ-
ence. Also in Eq. (4), Vl is the characteristic reduction potential
for the ion, that is, the potential that defines the minimum
energy necessary for a charge transfer between electrode and
ion.
Because the reaction rate make sense if, and only if, the elec-
trolytic cell is on a potential equal or greater that Vl, we multiply
the reaction rate by a conditional probability that appears inside
the first bracket in Eq. (4). This first bracket works like a switch
for the current (reduction process). This function mimics the
step function without loosing the continuous behavior. Vk is a
potential that localizes the half value of the function (a sigmoid
function) and is chosen because the trigger mechanism departs
from zero when V = Vl. The positive constant b that appears in
the formula is used to quantify the changes produced by the
sigmoid function.

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P.C.T. D’Ajello et al. / Electrochimica Acta 53 (2008) 3156–3165
just need to explicit their dependence on temperature and write
an expression for k.
of Eq. (4). There is no formal prescription to attain this goal.
¨
We will assume the naıve hypothesis that Ohm’s law is valid on
By inspecting Eq. (3) it is possible to see that the contribution
ofthediffusionconstantisalreadyincludedinthecurrentdensity
expression, obtained through the Fick’s second law. Because the
solvated ions has a behavior of Brownian particles we used the
Stoke–Einstein relation [28] to make explicit the temperature
dependence of the diffusion constant,
both sides of the electrode/liquid interface. It is not a problem if
the current that flows through the interface does not obey Ohm’s
law. We assert that before deposition, as the potential is turned
on, there is a charge arrangement in both sides of the interface
relating the electric field E, the resistivity ρ and the current den-
sity in the usual way, that is E = ρJ. Inside the electrode it reads
Es = ρsJs, and similarly in the liquid side, near the electrode sur-
face, we have El = ρlJl (the subscripts identify the medium, s for
silicon and l for the liquid).
ꢄ
ꢅ
R
1
D =
T,
(5)
N 6πμr
Because the electric charges on the electrode side induce an
equal amount of charge on the liquid side, we conjecture that the
current densities are proportional, i.e., Js ∝ Jl. Now, invoking
the classical image of a parallel plate capacitor of width δ and
capacitance CA, defined and charged in the liquid side, just on
the electrode surface, we write,
where μ is the viscosity of the liquid and r is the radius of the
Brownian particles.
Then, once the diffusion constant Dr is known at a particular
temperature Tr, we may write a simpler expression for it, that
is, D = AT, where the parameter A turns to be a constant given
by A = Dr/Tr. Thus, to consider situations where T lies around
the ambient temperature we use Dr = 1.0 × 10−5 cm2 s−1 and
Tr = 300 K to define A and,
E
Q V
RT
V
=
= CAρlδlJl
,
(9)
kBT
RT
D = 3.4 × 10−8T (cm2/s).
Also, the resistivity ρ in the electrolytic medium and the pop-
ulation of conduction electrons in the electrode can be put as a
function that depends uniquely on the temperature, as it will be
shown in the next paragraph.
Following the works of Rajendran et al. [29] and Villullas
et al. [30], for situations that evolve in the range 277–314 K, a
simple relation connecting the resistivity with the inverse of the
temperature could be used. However, this relation is only valid
for electrolytes with low concentration of ionic species, because
we know that the electrical resistivity depicts a nonlinear depen-
dence with the temperature at high electrolyte concentrations
[31–33]. We also know that the resistivity (conductivity) of a
complex mixture of electrolytes is not available by a simple
expression, however it is accepted as an inverse (linear) func-
tionoftemperatureforsystemsnearthe300 K [29–33]andunder
conditions of low electrolyte concentration. So after Rajendran
et al. [29], we assume,
(6)
where we have used the resistance in the form Rl = ρlδl/A.
To relate δlJl (whose dimension is molar charge per length
per second) to our selected parameters (N and D) we consider the
fluxofionsneartheelectrodesurface(Jl ∝ D(∂c/∂x))andassume
that δl(∂c/∂x) ∝ N and δlJl ∝ Dδl(∂c/∂x) ∝ DN. That means that
the concentration gradient in the solution, in the close proximity
of the interface, is proportional to the electronic density on the
electrode surface. Thus we propose,
E
V
= CAρlNDzF
.
(10)
kBT
RT
Using Eq. (10) we write again Eq. (4), that reads,
k = [1 + exp{b(V − Vk)}]−1
ꢃ
ꢄ
ꢅꢆ−1
zFCAρlND(V − Vl)
× 1 + exp
.
(11)
RT
This expression gives the reaction rate in terms of the tem-
perature apart from a last ingredient. Temperature, besides the
effects produced on D, N and ρ, as shown by Eqs. (6)–(8),
also affects the reaction rate through a change on the activa-
tion energy required for the charge transfer. In the present case
depicting a twofold contribution. One of these consists in the
self-evident distortion of the potential barrier after an increment
on the temperature. In Fig. 1 we demonstrate schematically the
concept. In this figure U and UI represent the magnitude of
the energy barriers at the reference temperature and at a higher
one, respectively. In Eq. (11) U is given by b(V − Vk) and
zFCAρlND(V − Vl), two terms with similar meaning that we
will represent by Q(V − Vl) for brevity. To make simpler our
explanation we will just examine the representative relation,
ꢄ
ꢅ
1
ρ =
(cm/S).
(7)
1.08 × 10−4T
To complete the list of physical parameters affected by the
temperature that enters in the description of the reaction rate, we
will write below the expression for the population of electrons in
the conduction band of a semiconductor. From current textbooks
[34] we find,
ꢄ
ꢅ
√
Eg
1
N = 4.83 × 1015 T3 exp
−
,
(8)
2kBT cm3
where Eg is the energy band gap and kB is the Boltzmann con-
stant.
ꢃ
ꢄ
ꢅꢆ
Given that we have identified and made explicit the rela-
tion between each physical variable and the temperature, we
should now conclude the task by relating ρ, N and D together,
in order to define the arbitrary parameter ␣ and the argument
of the exponential function that appears in the second bracket
Q(V − Vl)
k ∝ 1 + exp
,
(12)
RT
that corresponds to Eq. (11) with just one of the bracket in
an explicit form. It is enough to examine this generic bracket

P.C.T. D’Ajello et al. / Electrochimica Acta 53 (2008) 3156–3165
3159
of electrons are now functions of T as shown by Eqs. (6)–(8).
As a consequence, the current density, given by Eq. (3), is now
in the general formula I = I(cb, ω, T, V), as announced in the
introduction of the paper. ω is included to account for the scan
rate that relates the potential V to the time t.
3. Effect of temperature on the voltammograms
Through the algorithm given in the Appendix B we use the
expression,
ꢀ
ꢂ
∞
2
ꢁ
exp(−kt − exp(−λi t))
I = −Ista 1 − exp(−kt) + 2k
,
λ2i − k
i=1
(14)
Fig. 1. Double-well potential that represents the potential energy to be surmount
in a charge transfer “reaction” that imply the reduction of ionic specie on the
electrode surface. The sketch mimics the changes on the potential barrier by
effect of temperature (Ts > Tc) when we look the same region in the space of
parameters.
which generate all the theoretical current–potential curves
shown in this work.
Eq. (14) is identical to Eq. (3), but now the reaction rate k(V,
T) is specified by Eq. (13) and the time t(V, ω) is substituted by
a function of the potential and the scan rate through the relation
V = V0 + ωt.
because all the effects verified there have a similar effect on the
other. Assuming that Vl is a characteristic and constant potential
in Eq. (12), it follows that and increasing in T requires a lower
numerator (on the exponential argument) in order to that cor-
responds to Eq. (11) with just one of the bracket in an explicit
form. It is enough to maintain the same magnitude for k, and that
means we attain this magnitude for the reaction rate at a lower
value for the variable potential V. However we realize another
contribution coming from the temperature control, it redefines,
Vl, the potential that triggers the deposition process. To explain
this assertion we remember that, in the electrolytic solution, the
polarized water molecules encapsulate the ions by many solva-
tion shells. Thus, it is natural to assume a redox process requires
these shells to be broken or to be found in an unstable state
in order to allows permeability for electron transport. Thus an
increase on the temperature also increases the probability that
charges permeate the solvation shells, reducing the ion. In our
schematic description, depicted in Fig. 1, this effect is visualized
as a lowering of Vl, the potential yield that sets the beginning of
thereaction. Thetheoreticalmodelfollowsthesimplestprescrip-
tion, assuming that Vl changes with the inverse of temperature.
Once the range of variations on T is short we follow the current
dependence chosen for all parameters in this paper (except to N)
assuming in this case an inverse linear dependence with temper-
ature. Because all variations are computed in terms of a reference
temperature Tr, to take account of this effect we change Vl by
(Tr/T)Vl, in Eq. (11). As a consequence, the final expression for
the reaction rate reads,
Before continuing we must observe that in spite of our effort
to avoid arbitrary assumptions on the definition of the current
expressions, we are faced to our incapability to define a precise
value for the stationary concentration on the electrode surface,
cs, as function of the temperature as well as the impossibility to
define a precise value for the resistivity and the capacitance in
systems composed by complex electrolyte mixtures. We also
assume that δ goes with the inverse of temperature because
thermal agitation tends to reduce the region where the induced
dipoles are lined up to the electric field. So we wrote δ = δr(Tr/T),
with the subindex meaning the reference values. Thus, to eval-
uate the adequacy of our description we will proceed as follow.
We adjust the theoretical results to the experimental data at
T = 300 K, in order to fix these parameters. Once it is done, noth-
ingbuttemperatureischangedtoproducethecurves. In Table1it
is given the magnitude of all parameters used for T = Tr = 300 K.
We search for a theoretical approach suitable for the
description of the voltammograms. That means we are pursuing
for a qualitative description of the experimental profiles,
which could be achieved in terms of real variables and with a
minimum requirement of free parameters. In Fig. 2 we show
the current–potential profiles obtained from Eq. (14). All the
curves are defined by the same set of parameters, except for the
temperature. In this figure we see how the theoretical model
reveals the main effects produced by increments on the bath
temperature, namely the increase of the current peak yield and
the increase of the stationary current (represented by the plateau)
ꢃ
ꢄ
ꢅꢇꢆ−1
Tr
V − Vk
T
k = 1 + exp
b
Table 1
Values used to generate the voltammograms
ꢃ
ꢄ
ꢅꢆ−1
zFCAρND(V − Tr/T)Vl
cb = 26 mM
cs = 14 mM
Vk = 0.89 V
Vk(return) = 0.87 V
× 1 + exp
.
(13)
RT
δ = 3 × 10−2 cm
D = 1 × 10−5 cm2 s−1
b = 200 V−1
Vl = 0.91 V
ω = 10 mVs−1
In this formulation the reaction rate k (given by Eq. (13))
is a function of the temperature T and the potential V, given
that the resistivity, the diffusion coefficient and the population
ρCA = 1.79 × 10−6 cm F S−1
breturn = 32 V−1
N = 1.43 × 1010 cm−3

3160
P.C.T. D’Ajello et al. / Electrochimica Acta 53 (2008) 3156–3165
by effect of the temperature. For instance we could explain the
growth verified on the stationary current (the intensity of the cur-
rent plateaus) with a temperature increment as a consequence of
the increment on the diffusion constant and a reduction of the
depth of the depletion layer. However, the strong fluctuations
observed in the stationary current, particularly for the temper-
ature of 314 K, are not described by the model developed in
this work since they are a consequence of non-regular natural
convection movements inside the electrolytic cell, an effect not
included into the model yet. Despite some imperfections, the
general behavior depicted by the theoretical curves agrees quite
well with the experimental ones.
Also, according to the model, the changes on the nucleation
loops are related to the changes promoted by the temperature in
the magnitude of the diffusion constant, the conductivity of the
electrolytic solution and the way it re-scales the magnitude of
the potential that start the deposition process.
Fig. 2. Theoretical voltammograms obtained from Eq. (14). They represent sys-
tems, which are defined by the same parameters except temperature. To generate
the curves we used a 10 mV s−1 scan rate, 0.03 cm as the width of the stationary
diffusion layer and ρ, N and D as defined in the Table 1.
Another point to be discussed is related to the coupling
between the current peaks and the current plateaus. The inten-
sity of the current peaks depend on both, the magnitude of the
stationary current Ista and the magnitude of the function that
appears inside the brackets, in Eq. (14), which is defined by
the magnitude of the reaction rate k. Because Ista is defined
in terms of T, as it occurs to be with k and with the function
inside the brackets, the magnitude of the current peaks and that
of the current plateaus are proportional. This proportionality
could be confirmed comparing profiles (voltammograms) that
are normalized by two different procedures. If we normalize
each voltammogram by its current peak, we will see that the
plateaus merge on a same current value. On the other hand, if
we normalize each voltammogram by its current plateau value,
then the current peaks will attain equal magnitude. This behav-
ior, not explicitly shown here, is a common feature of theoretical
and experimental voltammograms and proves the proportional-
ity between peaks and plateaus. This benefit put forward by the
model, allows us to consider the relationship between the sta-
tionary currents measured through voltammograms and the ones
measured through current transients (current–time profiles). In
fact, we verified the coincidence among the stationary current
values obtained from the voltammograms (the plateau inten-
sity) and the stationary currents given by the current transients
(the current intensity after a long deposition time). This result,
evidenced by the experimental data and confirmed by the the-
oretical model, shows (see Eq. (3)) that Ista = −zFD(cb − cs)/δ
is a function that depends merely on temperature. Then, if the
current transient profiles are taken at just one temperature (no
matter what the deposition potential is), the stationary current
value depicted by the curves should coincide with the current
intensities shown by the voltammogram measured at the same
temperature. In Fig. 4 we show current transients obtained at dif-
ferent potential values, but at the same temperature (T = 289 K).
These transients displays stationary currents that merge to a cur-
rent value Ista that corresponds to the one defined by the plateau
in the voltammogram measured at the same temperature and
depicted in Fig. 3. The theoretical model gives us a result (not
shown) that is in close agreement with that conclusion, whose
merit is to emphasize the correspondence between both descrip-
on the voltammograms. Alterations on the nucleation loop and
on the potential that triggers the deposition current are also
verified. It is particularly apparent the shift, of the potential that
triggers deposition, to less negative values when temperature is
increased. All the effects identified in the voltammograms are
very often observed in experimental voltammograms.
Fig. 3 shows, similarly to Fig. 2, three voltammograms
obtained at three different temperatures. The experimental pro-
cedure to obtain the data is described in detail at the Appendix A.
The features in the nucleation loops, stationary current and shift
in the potential onset are easily observed and the similarities
between theoretical and experimental curves allow us to explain
the reasons that are behind every one of the changes introduced
Fig. 3. Experimental voltammograms obtained during electrochemical deposi-
tion of Co on n-silicon electrode. The electrolytic medium was formed by an
aqueous solution containing 26 mM of CoSO4 and 0.5 M of Na2SO4, and the
scan rate was 10 mV/s. The current profiles were obtained at different tempera-
tures as indicated in the figure.

P.C.T. D’Ajello et al. / Electrochimica Acta 53 (2008) 3156–3165
3161
with those shown in Figs. 2 and 3 through the corresponding
voltammograms.
4. Conclusions
In this paper we have improved the expression that defines the
reaction rate entering the description of the voltammograms and
the current transients produced during electrochemical depo-
sition processes. We have introduced the temperature as a
fundamental variable to define the kinetic (diffusion and reac-
tion) that evolves out of equilibrium conditions.
To make explicit the dependence of the reaction rate k with
temperature, it was necessary to develop a dimensional analysis
involving a set of parameters that depend explicitly on tem-
perature, namely the diffusion coefficient D, the capacitance
CA, the electrical resistivity in the electrolytic solution and the
population of electrons in the conduction band of the semicon-
ductor electrode. Because the dimensional analysis requires all
this parameters to define a non-dimensional factor entering on
the definition of the reaction rate k, we insist to represent the
diffusion coefficient as a function of temperature, even when
weakly affected by the temperature in the range considered.
In comparison to previous articles [1,2] we lost a formal ele-
gance of our early description, but we succeed for a current
expression gives in terms of real parameters. In addition, to offer
an analytical description of a complex phenomenon, the model
allows for a qualitative comparison of theoretical and exper-
imental data. Because the expression for the current–potential
profiles were made barely explicit, as function of physics param-
eters, we can hope that after major improvements, descriptions
like this could be used in the future as a tool to quantify ρ, D
and CA for ion migration (under diffusion control) in a great
number of electrolytic solutions and under diverse conditions of
temperature and ion concentration. Then electrochemistry will
offer a powerful analytical procedure to quantify electrical prop-
erties and transport coefficient that characterizes bio-chemical
systems.
Fig. 4. Experimental current–time curves. The transients were generated at three
different potential as indicated in the figure. The temperature of the samples
during the acquisition of the data was 289 K and we indicate the magnitude of
the current density at the peak of the profiles.
tions of the current profiles I(cb, T, t)and I(cb, T, V), prescribed
by the model.
To examine the profound relation between the current tran-
sients I(cb, T, t) and the voltammograms I(cb, T, V), we show
in Fig. 5 theoretical current transients taken at the same poten-
tial but at different temperatures. Once again, we verify that a
change in the temperature produces different stationary currents
as shown by the correspondent voltammograms. It is important
to stress that through the piece of the current profile shown in
Fig. 5 it was not possible to verify the exact stationary value for
the current as obtained by the voltammograms. However, if we
take these transients for longer times we certainly will verify
those values. A comparison between Figs. 4 and 5 allow us to
verify that in one case (in Fig. 4) the current peak intensity depict
different values because the currents were recorded under dif-
ferent potential whereas in Fig. 5 the reason for the difference
on the peak magnitudes is just due to the temperature differ-
ence among the current transients. A result in perfect agreement
Acknowledgements
The authors acknowledge research support from CAPES,
CNPq, FAPESC, and FINEP.
Appendix A. Experimental procedure
The experimental data used in this work were obtained with
a conventional three-electrode cell. The electrolyte employed
was aqueous solution containing 26 mM CoSO4 as the source
of metal ions, 500 mM of Na2SO4 as the supporting electrolyte,
with pH 4.5. The temperature of the electrochemical bath were
carefully controlled and fixed during the acquisition of each
curve (voltammograms and current transients). The working
electrode used in our experiments were silicon, obtained from
single side polished, technical grade (100) oriented Si wafers
n-doped (resistivity of 1–10 ꢀ cm). The electrical contact to the
working electrode was made through eutectic GaIn back con-
tact. A tape was used to mask off all the electrode except for a
Fig. 5. Theoretical current–time curves, generated by Eq. (14). The correspond-
ing reaction rate is computed from Eq. (13) and the temperatures are indicated
in the figure.

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circular area (0.478 cm2) of Si(1 0 0) surface which contact the
electrolyte. Prior to each electrochemical experiment, a fresh
electrode was prepared and its silicon surface was cleaned with
5% HF solution, then immediately transferred to the electro-
chemical cell. The Pt foil counter electrode was placed directly
opposite the working electrode. The potentials were measured
against a saturated calomel reference electrode (SCE) which was
placed as close as possible to the working electrode surface to
minimize the ohmic potential drop in the electrolyte. The elec-
trochemical experiments were performed in a dark chamber. All
the electrolytes, as well as the etching solutions used to clean the
samples prior to the electrochemical experiments, were prepared
from distilled deionized water with a resistivity of 18 ꢀ cm and
analytical grade reagents.
Appendix B. Program to compute voltammograms

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References
[12] B.R. Scharifker, G. Hills, Electrochim. Acta 28 (1983) 879.
[13] B.R. Sharifker, J. Mostany, J. Electroanal. Chem. 177 (1984) 13.
[14] M. Sluyters-Rehbach, J.H.O.J. Wijenberg, E. Bosco, J. Sluyters, J. Elec-
troanal. Chem. 236 (1987) 1.
[15] M.V. Mirkin, A.P. Nilov, J. Electroanal. Chem. 283 (1990) 35.
[16] L. Heerman, A. Tarallo, J. Electroanal. Chem. 470 (1999) 70.
[17] L. Heerman, A. Tarallo, Electrochem. Commun. 2 (2000) 85.
[18] P.C.T. D’Ajello, M.A. Fiori, A.A. Pasa, Z.G. Kipervaser, J. Electrochem.
Soc. 147 (2000) 4562.
[19] P.C.T. D’Ajello, M.L. Munford, A.A. Pasa, J. Chem. Phys. 111 (1999)
4267.
[20] S. Fletcher, J. Chem. Soc., Faraday Trans. 79 (1) (1983) 467.
[21] R.L. Deutscher, S. Fletcher, J. Electroanal. Chem. 239 (1988) 17.
[22] R.L. Deutscher, S. Fletcher, J. Electroanal. Chem. 277 (1990) 1.
[23] S. Fletcher, J. Electroanal. Chem. 215 (1986) 1.
[1] S. Wu, H.X. Ju, Y. Liu, Adv. Funct. Mater. 17 (2007) 585.
[2] B. Ballarin, M.C. Cassani, R. Mazzoni, E. Scavetta, D. Tonelli, Biosens.
Bioelectron. 22 (2007) 1317.
[3] M.R. Newton, K.A. Morey, Y. Zhang, R.J. Snow, M. Diwekar, J. Shi, H.S.
White, Nano Lett. 4 (2004) 875.
[4] A.M. Yu, F. Meiser, T. Cassagneau, F. Caruso, Nano Lett. 4 (2004) 177.
[5] N. Eliaz, M. Eliyahu, J. Biomed. Mater. Res. A 80A (2007) 621.
[6] M.E. Hyde, R.G. Compton, J. Electroanal. Chem. 549 (2003) 1.
[7] B.R. Sharifker, G.J. Hills, J. Electroanal. Chem. 130 (1981) 81.
[8] G.J. Hills, I. Montenegro, B.R. Scharifker, J. Appl. Electrochem. 10 (1980)
807.
[9] G.A. Gunawardena, G.J. Hills, I. Montenegro, Electrochim. Acta 23 (1978)
693.
[10] G. Gunawardena, G. Hills, I. Montengro, B. Scharifker, J. Electroanal.
Chem. 138 (1982) 225.
[11] L. Heerman, E. Matthijs, S. Langerlock, Electrochim. Acta 47 (2001) 905.
[24] A. Milchev, J. Electroanal. Chem. 457 (1998) 47.
[25] A. Milchev, J. Electroanal. Chem. 457 (1998) 35.
[26] P.C.T. D’Ajello, J. Electroanal. Chem. 573 (2004) 29.

P.C.T. D’Ajello et al. / Electrochimica Acta 53 (2008) 3156–3165
3165
[27] P.C.T. D’Ajello, A.Q. Schervenski, J. Electroanal. Chem. 573 (2004) 37.
[28] A. Einstein, Ann. der Phys. 17 (1905) 549.
[32] A. de Diego, A. Usobiaga, J.M. Madariaga, J. Electroanal. Chem. 446
(1998) 177.
[29] A. Rajendran, P. Neelamegam, Measurement 35 (2004) 59.
[30] H.M. Villullas, E.R. Gonzalez, J. Phys. Chem. B 109 (2005) 9166.
[31] J.O’.M. Bockris, A.K.N. Reddy, in: S.A. Reverte´ (Ed.), Electroqu´ımica
Moderna, Barcelona, 1980.
[33] A.V. Sharygin, I. Mokbel, C. Xiao, R.H. Wood, J. Phys. Chem. B 105
(2001) 105.
[34] C. Kittel, Introduction to Solid State Physics, Wiley & Sons, Inc.,
1996.